{"id":1894,"date":"2021-09-22T21:15:39","date_gmt":"2021-09-22T12:15:39","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1894"},"modified":"2021-09-22T21:15:39","modified_gmt":"2021-09-22T12:15:39","slug":"tbr201602","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr201602\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2016\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(xy\\) \u5e73\u9762\u306e\u76f4\u7dda \\(y = \\left( \\tan 2 \\theta \\right) x\\) \u3092 \\(\\ell\\) \u3068\u3059\u308b. \u305f\u3060\u3057, \\(0 \\lt \\theta \\lt \\dfrac{\\pi}{4}\\) \u3068\u3059\u308b. \u56f3\u3067\u793a\u3059\u3088\u3046\u306b, \u5186 \\(C _ 1 , C _ 2\\) \u3092\u4ee5\u4e0b\u306e (i) \uff5e (iv) \u3067\u5b9a\u3081\u308b.<\/p>\r\n

    \r\n

  1. (i)\u3000\u5186 \\(C _ 1\\) \u306f\u76f4\u7dda \\(\\ell\\) \u304a\u3088\u3073 \\(x\\) \u8ef8\u306e\u6b63\u306e\u90e8\u5206\u3068\u63a5\u3059\u308b.<\/p><\/li>\r\n

  2. (ii)\u3000\u5186 \\(C _ 1\\) \u306e\u4e2d\u5fc3\u306f\u7b2c \\(1\\) \u8c61\u9650\u306b\u3042\u308a, \u539f\u70b9 O \u304b\u3089\u4e2d\u5fc3\u307e\u3067\u306e\u8ddd\u96e2 \\(d _ 1\\) \u306f \\(\\sin 2 \\theta\\) \u3067\u3042\u308b.<\/p><\/li>\r\n

  3. (iii)\u3000\u5186 \\(C _ 2\\) \u306f\u76f4\u7dda \\(\\ell\\) , \\(x\\) \u8ef8\u306e\u6b63\u306e\u90e8\u5206, \u304a\u3088\u3073\u5186 \\(C _ 1\\) \u3068\u63a5\u3059\u308b.<\/p><\/li>\r\n

  4. (iv)\u3000\u5186 \\(C _ 2\\) \u306e\u4e2d\u5fc3\u306f\u7b2c \\(1\\) \u8c61\u9650\u306b\u3042\u308a, \u539f\u70b9 O \u304b\u3089\u4e2d\u5fc3\u307e\u3067\u306e\u8ddd\u96e2 \\(d _ 2\\) \u306f \\(d _ 1 \\gt d _ 2\\) \u3092\u6e80\u305f\u3059.<\/p><\/li>\r\n<\/ol>\r\n

    \u5186 \\(C _ 1\\) \u3068\u5186 \\(C _ 2\\) \u306e\u5171\u901a\u63a5\u7dda\u306e\u3046\u3061, \\(x\\) \u8ef8, \u76f4\u7dda \\(\\ell\\) \u3068\u7570\u306a\u308b\u76f4\u7dda\u3092 \\(m\\) \u3068\u3057, \u76f4\u7dda \\(m\\) \u3068\u76f4\u7dda \\(\\ell\\) , \\(x\\) \u8ef8\u3068\u306e\u4ea4\u70b9\u3092\u305d\u308c\u305e\u308c P , Q \u3068\u3059\u308b.<\/p>\r\n

      \r\n

    1. (1)<\/strong>\u3000\u5186 \\(C _ 1 , C _ 2\\) \u306e\u534a\u5f84\u3092 \\(\\sin \\theta , \\cos \\theta\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000\\(\\theta\\) \u304c \\(0 \\lt \\theta \\lt \\dfrac{\\pi}{4}\\) \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \u7dda\u5206 PQ \u306e\u9577\u3055\u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    3. (3)<\/strong>\u3000(2)<\/strong> \u306e\u6700\u5927\u5024\u3092\u4e0e\u3048\u308b \\(\\theta\\) \u306b\u3064\u3044\u3066\u76f4\u7dda \\(m\\) \u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n\"tbr20160201.svg\"\r\n

      \r\n\r\n

      \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n\"tbr20160202.svg\"\r\n

      (1)<\/strong><\/p>\r\n

      \u5186 \\(C_1 , C_2\\) \u306e\u534a\u5f84\u3092\u305d\u308c\u305e\u308c \\(r_1 , r_2\\) \u3068\u304a\u304f.
      \r\n\u5186 \\(C_1 , C_2\\) \u306e\u63a5\u70b9\u3092 R , \u305d\u308c\u305e\u308c\u306e\u4e2d\u5fc3\u3092 \\(\\text{O} {} _ 1 , \\text{O} {} _ 2\\) , \\(x\\) \u8ef8\u3068\u306e\u63a5\u70b9\u3092 \\(\\text{H} {} _ 1 , \\text{H} {} _ 2\\) \u3068\u304a\u304f.
      \r\n\\(d_1 = \\sin 2 \\theta\\) , \\(\\angle \\text{O} {} _ 1 \\text{OH} {} _ 1 = \\theta\\) \u306a\u306e\u3067\r\n\\[\r\nr_1 = \\sin 2 \\theta \\cdot \\sin \\theta = \\underline{2 \\sin^2 \\theta \\cos \\theta} \\ .\r\n\\]\r\n\\(d_2 = \\dfrac{r_2}{\\sin \\theta}\\) \u3067\u3042\u308a, \\(d_1 = d_2 +r_1 +r_2\\) \u306a\u306e\u3067\r\n\\[\\begin{gather}\r\n\\dfrac{r_2}{\\sin \\theta} +r_2 +2 \\sin^2 \\theta \\cos \\theta = 2 \\sin \\theta \\cos \\theta \\\\\r\n\\text{\u2234} \\quad r_2 = \\underline{\\dfrac{2 \\sin^2 \\theta \\cos \\theta ( 1 -\\sin \\theta )}{1 +\\sin \\theta}} \\ .\r\n\\end{gather}\\]\r\n

      (2)<\/strong><\/p>\r\n

      \\[\\begin{align}\r\n\\text{OR} & = \\dfrac{( 1 +\\sin \\theta ) r_2}{\\sin \\theta} \\\\\r\n& = 2 \\sin \\theta \\cos \\theta ( 1 -\\sin \\theta ) \\ .\r\n\\end{align}\\]\r\nR \u306f PQ \u306e\u4e2d\u70b9\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\text{PQ} & = 2 \\tan \\theta \\cdot \\text{OR} \\\\\r\n& = 4 \\sin ^2 \\theta ( 1 -\\sin \\theta ) \\ .\r\n\\end{align}\\]\r\n\\(x = \\sin \\theta\\) \u3068\u304a\u304f\u3068, \\(0 \\lt x \\lt \\dfrac{1}{\\sqrt{2}}\\) ... [1] .
      \r\n\\(f(x) = x^2 ( 1-x )\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(x) = 2x -3x^2 = x ( 2 -3x ) \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] \u306b\u304a\u3051\u308b \\(f(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} x & ( 0 ) & \\cdots & \\dfrac{2}{3} & \\cdots & \\left( \\dfrac{1}{\\sqrt{2}} \\right) \\\\ \\hline f'(x) & & + & 0 & - & \\\\ \\hline f(x) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3086\u3048\u306b, \\(f(x)\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\nf \\left( \\dfrac{2}{3} \\right) = \\dfrac{4}{9} \\cdot \\dfrac{1}{3} = \\dfrac{4}{27} \\ .\r\n\\]\r\n\u3088\u3063\u3066, PQ \u306e\u9577\u3055\u306e\u6700\u5927\u5024\u306f\r\n\\[\r\n4 \\cdot \\dfrac{4}{27} = \\underline{\\dfrac{16}{27}} \\ .\r\n\\]\r\n

      (3)<\/strong><\/p>\r\n

      \\(\\sin \\theta = \\dfrac{2}{3}\\) \u306e\u3068\u304d\r\n\\[\r\n\\cos \\theta = \\dfrac{\\sqrt{5}}{3} \\ , \\ \\tan \\theta = \\dfrac{2}{\\sqrt{5}} \\ .\r\n\\]\r\n\u76f4\u7dda \\(m\\) \u306f OR \u3068\u5782\u76f4\u306a\u306e\u3067, \u50be\u304d\u306f\r\n\\[\r\n\\tan \\left( \\theta +\\dfrac{\\pi}{2} \\right) = -\\dfrac{1}{\\tan \\theta} = -\\dfrac{\\sqrt{5}}{2} \\ .\r\n\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n\\text{OQ} & = \\dfrac{\\text{OR}}{\\cos \\theta} = 2 \\sin \\theta ( 1 -\\sin \\theta ) \\\\\r\n& = 2 \\cdot \\dfrac{2}{3} \\cdot \\dfrac{1}{3} = \\dfrac{4}{9} \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u3053\u306e\u3068\u304d\u306e \\(m\\) \u306e\u5f0f\u306f\r\n\\[\r\n\\underline{y = -\\dfrac{\\sqrt{5}}{2} \\left( x -\\dfrac{4}{9} \\right)} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u306e\u76f4\u7dda \\(y = \\left( \\tan 2 \\theta \\right) x\\) \u3092 \\(\\ell\\) \u3068\u3059\u308b. \u305f\u3060\u3057, \\(0 \\lt \\theta \\lt \\dfrac{\\pi}{4}\\) \u3068 … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[158],"tags":[144,162],"class_list":["post-1894","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2016","tag-tsukuba_r","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1894","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=1894"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1894\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1894"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1894"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1894"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}